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Chapter 5: Problem 64

What is a system of nonlinear equations? Provide an example with your description.

Short Answer

Expert verified
A system of nonlinear equations is a system that consists of at least one nonlinear equation. An example of a nonlinear system is the system of equations: (1) \(y = x^2 + 3\) and (2) \(y = -2x + 1\)

Step by step solution

01

Definition of Nonlinear Systems

A system of nonlinear equations is a set of equations in which at least one equation is not linear. A linear equation is any equation that can be written in the form Ax + By = C, where A, B, and C are coefficients and x and y are variables. If an equation in the system cannot be written in this form because it contains terms like \(x^2\), \(xy\), or \(\sqrt{x}\), etc., then the system is considered nonlinear.
02

Characteristics of Nonlinear Systems

In comparison to linear systems, nonlinear systems have a few distinct characteristics. For instance, they do not obey the superposition principle, meaning the effect of different causes in the system is not necessarily the sum of the effects when each cause is acting independently. Hence, these systems can respond in a more complex way to inputs, leading to phenomena such as chaos or bifurcations. Likewise, their graphical representation can have more varied shapes than the straight lines of linear systems.
03

Example of a Nonlinear System

Consider the following system of equations: (1) \(y = x^2 + 3\), which is a parabola, and (2) \(y = -2x + 1\), which is a straight line. This system is a nonlinear system because the first equation is not linear, even though the second one is. Both the equations can be graphed to see the solution of this system. This system of equations has two solutions, occurring where the parabola intersects the straight line, which can be found algebraically by setting the right-hand sides of the two equations equal to each other and solving for \(x\).

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Most popular questions from this chapter

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